Function not conforming to DCP ruleset

I have the following function
log(2log(1/(1-x))). Its second derivative is ( -1+log(1/(1-x)) )/( (x-1)log(1/(1-x)) )^2. This is non-negative only if 1<= log(1/(1-x)). I am unable to get it to conform to the DCP ruleset since there is an extra condition. I would be grateful for any help or relevant direction.

It’s neither convex nor concave over the entirety of its “natural” domain; Therefore, there is unlikely to be a DCP representation. But you’ll get a special virtual award if you find one.

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